This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
6
votes
1answer
119 views

How can mathematics work in wildly different set theories?

There are several set theories, e.g. ZFC and NF, which often have different axioms or are even outright contradictory. And yet most of other branches of mathematics, e.g. abstract algebra or topology, ...
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0answers
40 views

Is PA the most common foundation for arithmetic?

Are Peano Axioms the most common and widely accepted axiomatization of arithmetic, just as ZFC is the most common and widely accepted foundation for all of mathematics?
3
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0answers
36 views

What type of reasoning is employed when studying metalogic?

When studying, or doing any kind of reasoning about logic, what type of logic is used? Or how is the reasoning structured?
2
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2answers
62 views

How are image and pre-image different from range and domain respectively?

How are image and pre-image different from range and domain respectively, in Layman's terms (as simple as possible)? Are they basically just keywords that often indicate more nuanced subsets of the ...
-1
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0answers
51 views

Is this why proving Con(ZFC) is impossible? [duplicate]

I've stumbled upon this post on MathOverflow, and the poster has something interesting to say. In short: if we were to come up with a mathematical proof of the consistency of ZFC, we would be able ...
4
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2answers
381 views

Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
0
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0answers
23 views

Why is it impossible to prove absolute consistency of a theory falling prey to Godel's theorems?

Why is it impossible to prove absolute consistency of a theory T falling prey to Godel's theorems? I understand that a theory falling prey to Godel's second incompleteness theorem cannot prove its own ...
0
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1answer
18 views

Need an assistance with a specific step of a specific Division Algorithm proof

I'm trying to wrap my head around a Division Algorithm's proof. That is, Let $a, b \in \mathbb{Z}, a \neq 0$. Then there are unique $q,r \in \mathbb{Z}$ such that $b = qa + r, 0 \leq r < |a|$. ...
13
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2answers
811 views

Why doesn't this definition of natural numbers hold up in axiomatic set theory?

I was reading about older definitions of the natural numbers on Wikipedia here (in retrospect, not the best place to learn mathematics) and came across the following definition for the natural ...
0
votes
2answers
50 views

Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
0
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1answer
40 views

Why include equality in FOL for ZFC?

What are the pros and cons of working with first-order logic with equality for constructing ZFC, when all you have to do is make '$x=y$' a shorthand for: $$'\forall z [z \in x \Leftrightarrow z \in y] ...
0
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1answer
35 views

Any foundational theory of math falls prey to the incompleteness theorems - true or false?

I heard somewhere on the internet once something along the following lines: Any conceivable foundational theory of mathematics (be it ZFC or, if ZFC was found to be inconsistent, some modification ...
2
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1answer
52 views

Is isomorphism defined between large categories?

By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are ...
2
votes
1answer
33 views

Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of ...
1
vote
1answer
88 views

What book on Set Theory is best to understand motivation for axiomatization?

I am Master of Science in ICT, and I had always been in loved in math. On University we haven't been doing any Foundational Mathematics, the closest being Automata Theory and mention of Church-Turing ...
38
votes
3answers
4k views

Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
0
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0answers
46 views

Prereqisites for: Subsystems of second order arithmetic

As the title suggests, im wondering what the prerequisites for Simpsons book, Subsystems of... are? Unfortunately I cant find it in the preface. My background is a Bachelor in Philosophy and ...
1
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1answer
69 views

Alternatives to pure quantifier logic

Are there some alternatives for pure quantifier logic? Pure quantifier logic is axioms and rules of inference added to proposition logic to result first order logic. Are there other axioms that ...
5
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0answers
74 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
5
votes
2answers
344 views

Seeking a new, more natural definition of the cartesian product of sets

In "standard" set theory usually we have the definition $(a,b) := \{ \{ a \} , \{a,b\}\}$, see for example wikipedia for other similar ones. Then if we set for two sets $A,B$ $$ A\times B := \{ (a,b) ...
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votes
3answers
76 views

Divide by 0 alternative [closed]

Cutting to the chase. I know you can't divide by zero. And I have read a good few explications for this. And I am happy with this as a fact. BUT my question is based on this: X / N = A "should ...
3
votes
1answer
83 views

Logic without content? Takeuti and Zaring

I am reading the book "Introduction to Axiomatic Set Theory" by Takeuti and Zaring, and I wonder if I understand the "language of logic" from chapter 2 properly. They write: The language of our ...
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0answers
84 views

References about positivism

To found the current set theory, it has been necessary to remove some paradoxes like the well-known Russel paradox. It was thus necessary to clarify why things like $$\{x : x \notin x\}$$ can't ...
2
votes
1answer
39 views

Constructing natural numbers as lists of units (possible infinite objects)

I'm puzzled by this question, which is more about relation between two type theoretic approaches. Nevertheless, It can be shortened to the question : When it is correct (if ever) to construct ...
4
votes
2answers
55 views

Axioms of Trigonometry

On Wikipedia it gives a picture of all trigonometric functions of an angle laid atop the unit circle, 1. Obviously there are other trigonometric identities, but what I'm wondering is, does ...
3
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1answer
75 views

Godel's proof for dummies [closed]

Can someone give me as simple-a-proof as possible for Godel's Incompleteness Theorem? I'd love to understand it more.
7
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3answers
160 views

Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
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0answers
18 views

Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an ...
0
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6answers
66 views

How can I prove that if $a^7 = b^7$ then $a=b$, with $ a,b \in \mathbb{Z} $

I've tried with divisibility, meaning that since $a$ divides $a^7$, then $a$ divides $b^7$ and in the same way b divides $a^7$, but I can't seem to go further than this. What properties of the ...
0
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0answers
27 views

Understanding foundational terms: notions, objects and meta-objects

I am trying to take my problem solving skill to next level. It looks like It takes a lot of mathematical discipline. Here, This post buys me to get better at proof writing. So, I think is useful to ...
0
votes
1answer
35 views

Axioms of motion (Redei version)

I try to understand the axioms of motion in Redei's "Foundationd of Euclidean and Non-Euclidean Geometries, according to F. Klein" 1968 Redei gives as Axioms: Any motion is a one to one mapping of ...
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3answers
103 views

Are the assertions “$2 + 2$ equals $4$” and “$2 +2$ is $4$” identical

Are the assertions "$2 + 2$ equals $4$" and "$2 +2$ is $4$" identical? Or is this a linguistic, psychological or murky philosophical thing rather than a mathematical thing
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2answers
50 views

Foundations of Differential Calculus

In the preface of Foundations of Differential Calculus there's a section that says: Thus, if the quantity $x$ is given an increment $\omega$, so that it becomes $x + \omega$, its square $x^2$ ...
10
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2answers
551 views

What does “Mathematics of Computation” mean?

I visited this link: http://www.ams.org/journals/mcom/1950-04-030/S0025-5718-50-99474-9/ And I a bit confused by its title "Mathematics of Computation". I am not a native English speaker. Could ...
4
votes
1answer
94 views

Will Homotopy Type Theory ever be as accessible as traditional Set Theory?

At the moment, Homotopy Type Theory is barely accessible to undergraduates, and only the most advanced or most gifted could have a decent chance of grasping it at a workable level without mountains of ...
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0answers
52 views

Yoneda Lemma and foundations

Yoneda Lemma says that, for a locally small category $\mathcal{C}$, an object $A$ in $\mathcal{C}$ and a functor $F:\mathcal{C}^{op}\to \textbf{Set}$, the natural transformations $[Hom(-,A):F]$ is in ...
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vote
3answers
47 views

Bijective map on $(\Bbb N \times\Bbb N)/R$

I'm not sure how to tackle this problem. Consider the equivalence relation $R$ on $\Bbb N \times\Bbb N$ given by : $$(a, b)R(c, d) \iff a + d = b + c$$ (i) Show that $R$ is an equivalence ...
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3answers
134 views

Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
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2answers
76 views

What is the denial of a statement in logic math?

I'm trying to get the hang of denials in logic in math. I would like to use these two examples: "Some people are honest and some people are not honest. (All people)" "No one loves everybody. (All ...
0
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2answers
201 views

Can one use the Hilbert-Ackermann Consistency Theorem to prove the consistency of $PRA$?

In his textbook Mathematical Logic, Shoenfield states the Hilbert-Ackermann Consistency Theorem as follows: "Consistency Theorem (Hilbert-Ackermann): An open theory $T$ is inconsistent iff there is ...
3
votes
1answer
244 views

Theorems not Formulable in Set Theory

Several sites I have been reading say that set theory is a good foundation for mathematics because virtually every theorem can be cast into a theorem in set theory. What is an example of a theorem ...
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4answers
699 views

Continuum Hypothesis in formalized language. [closed]

The Continuum Hypothesis was advanced by Georg Cantor in 1878, before that Zermelo–Fraenkel set theory was stablished. "There is no set whose cardinality is strictly between that of the integers and ...
0
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0answers
52 views

Creating mathematics vs building houses

I found the following quote in the book "Calculus" by Michael Spivak. (At the first page of Part 5,Epilogue, where he will discuss fields, construction of the real number, and uniqueness of the real ...
1
vote
1answer
95 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
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0answers
41 views

Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number ...
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vote
2answers
56 views

Can the Recursion Theorem be proved in Peano Arithmetic?

A recursive function on $\mathbb{N}$ can be defined as follows: Given an element $a \in \mathbb{N}$ and a function $f:\mathbb{N}\rightarrow\mathbb{N}$, we can define a function ...
6
votes
1answer
82 views

Prob. 6, Chapter 1 in Baby Rudin [duplicate]

Here's problem $6$ in Chapter $1$ in the book Principles of Mathematical Analysis by Walter Rudin, $3$rd edition: Fix a real number $b$, such that $b > 1$. $(a)$ If $m, n, p, q$ are integers, ...
0
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1answer
29 views

Step 6, Appendix to Chapter 1 in Baby Rudin: How to show that if $\alpha > 0^*$ and $\beta > 0^*$, then $\alpha \beta > 0^*$?

I'm reading Appendix to Chapter 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition. In this appendix, Rudin gives a proof of Theorem 1.19 by constructing $\mathbb{R}$ from ...
6
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1answer
93 views

How can one prove the axiom of collection in ZFC without using the axiom of foundation?

Say I want to prove the axiom(s) of collection from the axiom(s) of replacement. If you have the axiom of foundation, then you can use Scott's trick to do this. But suppose I'm working in a context ...